Turing Pattern Generator

Governing Equation and Parameters

The pattern generation is based on the reaction-diffusion model described by the following equations:

du/dt = Du * ∇²u - uv² + F * (1 - u)
dv/dt = Dv * ∇²v + uv² - (F + k) * v

Here, u and v represent the concentration of two chemicals over time and space.

• Du (Diffusion rate of u): Controls the spread of the chemical u. Unit: cm²/s
• Dv (Diffusion rate of v): Controls the spread of the chemical v. Unit: cm²/s
• F (Feed rate): Controls the rate at which u is supplied to the system. Unit: 1/s
• k (Kill rate): Controls the rate at which v is removed from the system. Unit: 1/s

Changing these parameters affects the pattern formation as follows:

• Increasing Du or Dv generally smoothens the pattern, making it more diffusive.
• Higher F values speed up the pattern formation, resulting in smaller and more numerous features.
• Higher k values tend to create more complex and denser patterns.

References

This project uses the Gray-Scott Model of a Reaction-Diffusion System for pattern formation, originally developed and described by Katharina Käfer and Mirjam Schulz. The model is a fundamental system in reaction-diffusion studies and is widely used to simulate natural patterns.

For more detailed information on the Gray-Scott model, refer to their work:

• Katharina Käfer and Mirjam Schulz, "Gray-Scott Model of a Reaction-Diffusion System," available online at this link.